In the last 60 years conventional solid and electrolyte gating allowed sizable modulations of the surface carrier concentration in metallic superconductors resulting in tuning their conductivity and changing their critical temperature. Recent conventional gating experiments on superconducting metal nano-structures showed full suppression of the critical current without variations of the normal state resistance and the critical temperature. These results still miss a microscopic explanation. In this article, we show a complete set of gating experiments on Ti-based superconducting Dayem bridges and a suggested classical thermodynamic model which seems to account for several of our experimental findings. In particular, zero-bias resistance and critical current IC measurements highlight the following: the suppression of IC with both polarities of gate voltage, the surface nature of the effect, the critical temperature independence from the electric field and the gate-induced growth of a sub-gap dissipative component. In addition, the temperature dependence of the Josephson critical current seems to show the transition from the ballistic Kulik-Omelyanchuck behavior to the Ambegaokar-Baratoff tunnel-like characteristic by increasing the electric field. Furthermore, the IC suppression persists in the presence of sizeable perpendicular-to-plane magnetic fields. We propose a classical thermodynamic model able to describe some of the experimental observations of the present and previous works. Above all, the model grabs the bipolar electric field induced suppression of IC and the emergence of a sub-gap dissipative component near full suppression of the supercurrent. Finally, applications employing the discussed effect are proposed.
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